active and passive filters

Please have a look on this document. My question is about the text in maroon on page #3 where it says, "The cutoff frequency fc is the frequency above or below which the power of the circuit, which is most often a filter, is ½ the input power...".

At the critical frequency output voltage is 70.7% of the input voltage in a RC filter. What does it mean where the text says, "The cutoff frequency fc is the frequency above or below which the power of the circuit, which is most often a filter, is ½ the input power." Does it mean that at the critical frequency output power is half of input power in a RC circuit? I assume this is the case. Then, it says, "Since P = V^2/R, half the power is proportional to sqrt(1/2) the voltage." How is power proportional to sqrt(1/2) the input voltage?

Let's say you have a voltage Vo. How much power is that signal dumping into a resistive load R? Call that Po.

In terms of Vo, what would the voltage need to be in order to dump Po/2 into that same resistive load, R?

The definition of their cutoff frequency is a bit too simplistic, especially since the title of the thread includes active filters.

A much better description of the cutoff frequency of a filter is the frequency at which the power delivered to a resistive load becomes half of what it nominally is in the passband. This way, the definition is unamgiguous even if the filter has a loss or gain throughout the passband.